Weak compactness techniques for
coagulation equations.
I. The discrete coagulation equation
Philippe Laurençot
Institut de Mathématiques de Toulouse
25.07.2013
(IMT)
25.07.2013
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Coagulation/coalescence ...
Coagulation : growth of clusters by successive merging of smaller
clusters.
Exemples : polymers, droplets, animals, ...
Binary coagulation: two clusters merge into one.
Coagulation
(IMT)
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Representations
DISCRETE: Pi = particle with size/volume i, i ∈ N \ {0}
(polymers, animals)
Basic reaction:
Pi + Pj −→ Pi+j ,
i ≥ 1, j ≥ 1.
CONTINUOUS: Px = particle with size/volume x, i ∈ (0, ∞)
(droplets)
Basic reaction:
Px + Py −→ Px+y ,
(IMT)
x > 0, y > 0.
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Outline
1
The discrete Smoluchowski coagulation equation
2
Existence: Bounded coagulation kernels
3
Existence: Unbounded coagulation kernels
(IMT)
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The discrete Smoluchowski coagulation equation
Outline
1
The discrete Smoluchowski coagulation equation
2
Existence: Bounded coagulation kernels
3
Existence: Unbounded coagulation kernels
(IMT)
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The discrete Smoluchowski coagulation equation
Discrete coagulation equation
Pi = particle with size/volume i, i ∈ N \ {0}
Basic reaction:
Pi + Pj −→ Pi+j ,
i ≥ 1, j ≥ 1.
Conservation of matter & decrease of the number of particles
Coagulation
(IMT)
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The discrete Smoluchowski coagulation equation
Discrete coagulation equation
Mean-field model: dynamics of the size distribution function
f = (fi )i≥1 with the density fi of particles of size i ≥ 1.
Temporal variation of fi = [GAIN(Pi )] − [LOSS(Pi )],
[GAIN(Pi )] = “newly created particles Pi by merging of smaller
particles” (i > 1)
Pj + Pi−j −→ Pi ,
1 ≤ j ≤ i − 1.
[LOSS(Pi )] = “disappearance of particles Pi resulting from their
merging with another particle”.
Pi + Pj −→ Pi+j ,
Constitutive relation:
(IMT)
j ≥ 1.
Pi + Pj −→ Pi+j ∼ K (i, j) fi fj
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The discrete Smoluchowski coagulation equation
Discrete coagulation equation
i =1:

df1




dt














(IMT)
= −[LOSS(P1 )]
∞
X
= −
P1 + Pj −→ Pj+1
j=1
∞
X
= −
K (1, j) f1 fj .
j=1
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The discrete Smoluchowski coagulation equation
Discrete coagulation equation
i ≥2:

dfi




dt














(IMT)
= [GAIN(Pi )] − [LOSS(Pi )]
i−1
=
=
P 1 X
Pj + Pi−j −→ Pi − ∞
j=1 Pi + Pj −→ Pi+j
2
1
2
j=1
i−1
X
j=1
K (j, i − j) fj fi−j −
∞
X
K (i, j) fi fj .
j=1
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The discrete Smoluchowski coagulation equation
Discrete coagulation equation
i =1:
∞
X
df1
=−
K (1, j) f1 fj ,
dt
j=1
i ≥2:
i−1
∞
j=1
j=1
X
dfi
1X
=
K (j, i − j) fj fi−j −
K (i, j) fi fj .
dt
2
Coagulation kernel K (i, j) : K (i, j) = K (j, i) ≥ 0.
(IMT)
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The discrete Smoluchowski coagulation equation
Coagulation kernel K (i, j) : examples
K (i, j) =
i
1/3
+j
1/3
1
1
+
i 1/3 j 1/3
(Smoluchowski, 1916)
K (i, j) = 2
K (i, j) = (ai + b)(aj + b) , a > 0 , b ≥ 0
3
K (i, j) =
i 1/3 + j 1/3
2 1/3
K (i, j) =
i 1/3 + j 1/3
− j 1/3 i
K (i, j) = i α j β + i β j α ,
(IMT)
(Stockmayer , 1943)
α ≤ 1,β ≤ 1.
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The discrete Smoluchowski coagulation equation
Conservation of matter
Formally, if (ϑi )i≥1 is a sequence of real numbers, it follows from the
Fubini-Tonelli theorem that
∞
∞
d X
1 X
ϑ i fi =
ϑi+j − ϑi − ϑj K (i, j) fi fj .
dt
2
i=1
i,j=1
Choosing ϑi = i gives
∞
d X
ifi = 0
dt
−→
conservation of matter .
i=1
Remark: one needs to check that the Fubini-Tonelli theorem can be
applied.
(IMT)
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The discrete Smoluchowski coagulation equation
Discrete coagulation equation
∞
X
df1
i =1:
=−
K (1, j) f1 fj ,
dt
j=1
dfi
1
i ≥2:
=
dt
2
i−1
X
K (j, i − j) fj fi−j −
j=1
∞
X
K (i, j) fi fj ,
j=1
Coupled system of countably many ordinary differential equations:
For which coagulation kernels K (i, j) and initial data f in = fiin i≥1
does this system have solutions? Time span?
fiin ≥ 0, i ≥ 1, =⇒ fi (t) ≥ 0, i ≥ 1?
P in
If fiin ≥ 0, i ≥ 1, and
ifi < ∞, do we have
∞
X
i=1
(IMT)
ifi (t) < ∞
and
∞
X
i=1
ifi (t) =
∞
X
ifiin ?
i=1
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The discrete Smoluchowski coagulation equation
Explicit solutions K (i, j) = 1
K (i, j) = 1:
fi (t) =
2
2+t
2 t
2+t
i−1
,
i ≥1,
t ≥0.
Total mass:
∞
X
ifi (t) = 1 −→ conservation of mass .
i=1
(IMT)
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The discrete Smoluchowski coagulation equation
Explicit solutions K (i, j) = ij
K (i, j) = ij:
fi (t) =


i i−3

i−1 −it

e

 (i − 1)! t
if t ∈ [0, 1],



f (1)

 i
t
if t ∈ [1, +∞).
Total mass:
∞
X
i=1
1
ifi (t) = min 1,
−→ non-conservation of mass .
t
GELATION PHENOMENON/RUNAWAY GROWTH
(IMT)
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The discrete Smoluchowski coagulation equation
K (i, j) = ij
1.1
Total Density
1.0
0.9
0.8
N= 50
N= 150
N= 300
N= 500
N=
0.7
8
0.6
0.5
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time
(IMT)
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The discrete Smoluchowski coagulation equation
Explicit solutions
K (i, j) = 1: Other explicit solutions can be computed for specific
initial data (conservation of mass).
K (i, j) = i + j: Some explicit solutions can be computed, again for
specific initial data (conservation of mass).
K (i, j) = (ij)α , α ∈ (1/2, 1): Existence of solutions of the form
f in
fi (t) = i
,
1+t
i ≥1,
t ≥0,
with
∞
X
ifiin < ∞ .
i=1
−→ non-conservation of mass.
(IMT)
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The discrete Smoluchowski coagulation equation
K (i, j) = (ij)α , α ∈ (1/2, 1]
1.1
1.1
1.0
1.0
Total Density
0.8
N= 50
N= 150
N= 300
N= 500
N=
0.7
0.6
0.8
N= 50
N= 150
N= 300
N= 500
N= 800
N= 1000
0.7
0.6
0.5
8
Total Density
0.9
0.9
0.5
0.4
0
0.2
0.4
0.6
0.8
Time
1.0
1.2
1.4
1.6
0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
Time
Figure : K (i, j) = ij (left), K (i, j) = (ij)0.8 (right)
(IMT)
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Existence: Bounded coagulation kernels
Outline
1
The discrete Smoluchowski coagulation equation
2
Existence: Bounded coagulation kernels
3
Existence: Unbounded coagulation kernels
(IMT)
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Existence: Bounded coagulation kernels
Existence
Differential equation in an infinite dimensional Banach space
Theorem
Let (E, k.k) be a Banach space, G : E → E a locally Lipschitz
continuous function, and f in ∈ E. There is a unique maximal solution
f ∈ C 1 ([0, Tm ); E) to the differential equation
df
(t) = G(f (t)) ,
dt
t ∈ [0, Tm ),
f (0) = f in ,
and:
Tm = ∞
OR
Tm < ∞ and
lim kf (t)k = ∞ .
t→Tm
Consequence of the contraction mapping principle.
(IMT)
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Existence: Bounded coagulation kernels
Existence
with f = (fi )i≥1
df
= G(f ) , f (0) = f in = (fiin )i≥1 ,
dt
and G(f ) = (Gi (f ))i≥1 given by
∞
X
G1 (f ) = −
K (1, j) f1 fj ,
j=1
Gi (f ) =
i−1
∞
j=1
j=1
X
1X
K (j, i − j) fj fi−j −
K (i, j) fi fj ,
2
i ≥2.
Question: choice of a suitable functional setting.
(IMT)
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Existence: Bounded coagulation kernels
Existence
(
`1µ
:=
f = (fi )i≥1 : kf k1,µ =
∞
X
)
µ
i |fi | < ∞
,
µ∈R.
i=1
If
K (i, j) ≤ κ ,
i, j ≥ 1 ,
G : `10 → `10 is locally Lipschitz continuous (`10 ,→ `∞ ),
G is quasi-positive: if i ≥ 1 and f = (fj )j≥1 satisfies fj ≥ 0 pour
j ≥ 1 et fi = 0, then Gi (f ) ≥ 0,
∞
X
i=1
(IMT)
Gi (f ) = −
∞
X
K (i, j) fi fj .
i,j=1
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Existence: Bounded coagulation kernels
Existence
Theorem
If
K (i, j) ≤ κ ,
i, j ≥ 1 ,
and
f in = (fiin )i≥1 ∈ `10 ,
fiin ≥ 0 , i ≥ 1 ,
there is a unique global solution f = (fi )i≥1 ∈ C 1 ([0, ∞); `10 ) such that
fi (t) ≥ 0 for all t ≥ 0 and i ≥ 1. Furthermore, if f in ∈ `11 , then
f (t) ∈ `11 and
∞
X
i=1
ifi (t) =
∞
X
ifiin ,
t ≥ 0.
i=1
Boundedness of K −→ local Lipschitz continuity.
Non-negativity −→ global existence.
Conservation of mass −→ Fubini-Tonelli theorem.
(IMT)
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Existence: Bounded coagulation kernels
Decay of the total number of clusters
Assume further that
K (i, j) ≥ κ0 > 0 ,
Then
kf (t)k1,0 ≤
Indeed,
i, j ≥ 1 .
kf in k1,0
,
1 + κ0 kf in k1,0 t
t ≥0.
∞
X
d
kf k1,0 = −
K (i, j) fi fj ≤ −κ0 kf k21,0 .
dt
i,j=1
(IMT)
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Existence: Unbounded coagulation kernels
Outline
1
The discrete Smoluchowski coagulation equation
2
Existence: Bounded coagulation kernels
3
Existence: Unbounded coagulation kernels
(IMT)
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Existence: Unbounded coagulation kernels
Unbounded coagulation kernels
The previous approach does not seem to extend to unbounded
coagulation kernels −→ compactness method.
1
Build a sequence of approximations of the original problem:
which depends on a parameter N ≥ 1,
for which the existence of a solution is “easy” to show,
and “converges” to the original problem as N → ∞.
2
Derive estimates which are independent of N ≥ 1 and guarantee
compactness with respect to the size variable i and the time
variable t.
3
Show convergence as N → ∞.
(IMT)
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Existence: Unbounded coagulation kernels
Approximation
Let K be a coagulation kernel and consider f in = (fiin )i≥1 ∈ `11 satisfying
fiin ≥ 0 , i ≥ 1 ,
et
%=
∞
X
ifiin < ∞ .
i=1
“Truncation” parameter N ≥ 1. Define
K N (i, j) = min {K (i, j), N} ≤ N ,
i, j ≥ 1 .
According to the previous theorem, there exists a unique solution
f N = (fiN )i≥1 ∈ C 1 ([0, ∞); `10 ) ∩ L∞ (0, ∞; `11 )
to the DSCE with coagulation kernel K N and
fiN (t) ≥ 0 , i ≥ 1 ,
and
∞
X
i=1
(IMT)
ifiN (t) = % :=
∞
X
ifiin ,
t ≥ 0.
i=1
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Existence: Unbounded coagulation kernels
The limit N → ∞
∞
i−1
X
dfiN
1X N
N
K N (i, j) fjN , i ≥ 1 .
K (j, i − j) fjN fi−j
− fiN
=
dt
2
j=1
j=1
We have to identify the following limits:
fiN ,
∞
X
j=1
∞
X
dfiN
, i ≥ 1,
dt
as N → ∞ ,
K N (i, j) fjN as N → ∞ ,
j fjN as N → ∞ .
j=1
(IMT)
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Existence: Unbounded coagulation kernels
Compactness
Estimates:
fiN (t) ≥ 0 , i ≥ 1 ,
and
∞
X
ifiN (t) = % :=
i=1
∞
X
ifiin ,
t ≥ 0.
i=1
which implies that (fiN )N≥1 is bounded in L∞ (0, ∞) for each i ≥ 1.
Assume further that
K (i, j) ≤ Aij ,
Then
df N i
(t) ≤ 2A %2 ,
dt
(IMT)
i, j ≥ 1 .
t ≥ 0.
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Existence: Unbounded coagulation kernels
Compactness
Summarizing, we have proved that, if
K (i, j) ≤ Aij ,
i, j ≥ 1 ,
0 ≤ fiN (t) ≤ % for N ≥ 1, i ≥ 1, and t ≥ 0.
N
f (t) − f N (s) ≤ 2A %2 |t − s| for N ≥ 1, i ≥ 1, and t ≥ 0.
i
i
(IMT)
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Existence: Unbounded coagulation kernels
The Arzelà-Ascoli theorem
Uniform convergence of a sequence of continuous functions:
Theorem
Let T > 0 and (gn )n≥1 be a sequence of continuous functions from
[0, T ] to R enjoying the two properties:
there is C > 0 for which |gn (t)| ≤ C for all n ≥ 1 and t ∈ [0, T ],
Equicontinuity: there exists a continuous function
ω : [0, ∞) → [0, ∞) with ω(0) = 0 such that
|gn (t) − gn (s)| ≤ ω(|t − s|) ,
n ≥ 1 , (t, s) ∈ [0, T ]2 .
Then (gn ) is relatively compact in C([0, T ]).
(IMT)
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Existence: Unbounded coagulation kernels
Limit of (f N ) as N → ∞
Consequently, if
K (i, j) ≤ Aij ,
i, j ≥ 1 ,
0 ≤ fiN (t) ≤ % for N ≥ 1, i ≥ 1, and t ≥ 0,
N
f (t) − f N (s) ≤ 2A %2 |t − s| for N ≥ 1, i ≥ 1, and t ≥ 0,
i
i
Using the Arzelà-Ascoli theorem along with a diagonal process
give the existence of a sequence (Nk )k ≥1 , Nk → ∞, and a
sequence of continuous functions f = (fi )i≥1 such that
lim sup fiNk (t) − fi (t) = 0
k →∞ t∈[0,T ]
for all T > 0 and i ≥ 1,
f ∈ L∞ (0, ∞; `11 ) with kf (t)k1,1 ≤ kf in k1,1 for t ≥ 0 (Fatou lemma).
(IMT)
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Existence: Unbounded coagulation kernels
Limit of
P
K N (i, j) fjN
Fix i ≥ 1. Then, for 1 ≤ J ≤ N/Ai,
X
∞ N
N
K
(i,
j)
f
−
K
(i,
j)
f
j
j
j=1


J J X
X
N
≤
K (i, j) fjN − K (i, j) fj =
K (i, j) fjN − fj −→ 0
j=1
+
∞
X

K N (i, j) fjN ≤ Ai
j=J+1
+
∞
X
j=J+1
(IMT)
N→∞
j=1
∞
X

jfjN −→?
j=J+1

K (i, j) fj
≤ Ai
∞
X
j=J+1

jfj −→ 0
J→∞
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Existence: Unbounded coagulation kernels
Limit of
P
K N (i, j) fjN
Assume further that K (i, j) ≤ Aα (ij)α , i, j ≥ 1, for some α ∈ [0, 1).
Fix i ≥ 1. Then, for 1 ≤ J ≤ N/Ai,
X
∞
N
N
K (i, j) fj − K (i, j) fj j=1
J X
N
≤
−→ 0
K (i, j) fj − K (i, j) fj N→∞
j=1
+
∞
X

K N (i, j) fjN ≤ Aα i α
j=J+1
+
∞
X
j=J+1
(IMT)

∞
X
j α fjN ≤
j=J+1

K (i, j) fj
≤ Ai
∞
X
j=J+1
%
(J + 1)1−α
−→ 0
J→∞

jfj −→ 0
J→∞
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Existence: Unbounded coagulation kernels
Limit of
P
K N (i, j) fjN
Assume further that K (i, j) ≤ Aα (ij)α , i, j ≥ 1, for some α ∈ [0, 1).
Fix i ≥ 1and T > 0. Then,
∞
X N
Nk
k
K (i, j) fj (t) − K (i, j) fj (t) = 0 .
lim sup k →∞ t∈[0,T ] j=1
Recall that
lim sup fiNk (t) − fi (t) = 0 .
k →∞ t∈[0,T ]
Question: dfiN /dt?
(IMT)
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Existence: Unbounded coagulation kernels
Existence
Theorem
Assume that K (i, j) ≤ Aα (ij)α , i, j ≥ 1, for some α ∈ [0, 1) and consider
f in = (fiin )i≥1 such that fiin ≥ 0, i ≥ 1. Then there is
f = (fi )i≥1 ∈ C([0, ∞); `10 ) ∩ L∞ (0, ∞, `11 )
such that, for i ≥ 1 and t > 0, (weak formulation)


Z t
i−1
∞
X
X
1
K (j, i − j)fj (s)fi−j (s) − fi (s)
K (i, j)fj (s) ds .
fi (t) = fiin +
2
0
j=1
j=1
Furthermore, kf (t)k1,1 ≤ kf in k1,1 for t ≥ 0.
Remark. Weak formulation + regularity of f −→ fi ∈ C 1 ([0, ∞), i ≥ 1.
(IMT)
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Existence: Unbounded coagulation kernels
Comments
Existence for unbounded kernels.
Non-increase of total mass cannot be improved to conservation of
mass in general.
Uniqueness and continuous dependence have to be proved
separately (6= semigroup approach).
If K (i, j) ≤ Aα (ij)α for some α ∈ [0, 1] and f in ∈ `12α , then
uniqueness (and existence) of weak solutions which satisfy
additionally f ∈ L1 (0, T ; `12α ) for all T > 0.
Weak stability: if (f N )N≥1 is a sequence of solutions to the DSCE
such that f N,in −→ f in in `10 , there is a subsequence of (f N )N≥1
which converges to a solution of the DSCE with initial condition f in .
K (i, j) = i + j and K (i, j) = ij are excluded in the previous theorem.
(IMT)
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Existence: Unbounded coagulation kernels
Conservation of mass
Theorem
Assume that K (i, j) ≤ B(i + j), i, j ≥ 1, and consider f in = (fiin )i≥1 such
that fiin ≥ 0, i ≥ 1. If
f = (fi )i≥1 ∈ C([0, ∞); `10 ) ∩ L∞ (0, ∞, `11 )
is a weak solution to the DSCE, then
kf (t)k1,1 = kf in k1,1 ,
(IMT)
t ≥0.
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Existence: Unbounded coagulation kernels
Extensions
If K (i, j) ≤ B(i + j), i, j ≥ 1, existence of a mass-conserving weak
solution.
If K (i, j) = ri rj , i, j ≥ 1, existence of a weak solution whatever the
growth of ri .
If K (i, j) ≥ i α + j α , i, j ≥ 1, for some α > 1, non-existence.
(IMT)
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Existence: Unbounded coagulation kernels
Helly’s selection principle
Pointwise convergence of a sequence of functions:
Theorem
Let T > 0 and (gn )n≥1 be a sequence of functions from [0, T ] to R
enjoying the two properties:
there is C1 > 0 such that |gn (t)| ≤ C1 for all n ≥ 1 and t ∈ [0, T ],
Bounded Variation: there is C2 > 0 such that, for all n ≥ 1,
m ≥ 1 et 0 = t0 < t1 < . . . < tm−1 < tm = T , there holds
m
X
|gn (tl ) − gn (tl−1 )| ≤ C2 .
l=1
Then there are a subsequence (gnp )p≥1 of (gn ) and a bounded
function g such that gnp (t) −→ g(t) for all t ∈ [0, T ].
(IMT)
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